This is a GR question which is relevant to planetary orbits.
Is it correct to say that the Schwarzschild solution is the GR solution to the gravitational field around a point mass?
If we apply the Schwarzschild solution to the sun, we can treat the sun as a point mass. If we do this, we essentially recover Newton’s law since the velocity of the orbit is so slow compared to the speed of light?
If the speeds are very slow, the Schwarzschild solution (I.e. the metric tensor) is approximately \[ds^2\approx -\phi dt^2 + dx^2 + dy^2 + dz^2\]where \[\phi = c^2-\frac{2GM}{r}\] with \(M\) the mass of the sun and \[r^2=x^2+y^2+z^2.\]The geodesics parameterized by t give the equations of motion for \(x,y,z\) as \[\frac{d^2x}{dt^2}=\frac{\partial}{\partial x}\]and likewise for \(y\) and \(z\). This is exactly the equations of motion from Newton’s law of gravity due to the gravitational field of a point mass.
Is this more or less correct?
Diffgeometer1
npub1hd…k9se4
2025-07-17 20:15:50 UTC
in reply to nevent1q…sdfy
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"content": "This is a GR question which is relevant to planetary orbits. \n\nIs it correct to say that the Schwarzschild solution is the GR solution to the gravitational field around a point mass? \n\nIf we apply the Schwarzschild solution to the sun, we can treat the sun as a point mass. If we do this, we essentially recover Newton’s law since the velocity of the orbit is so slow compared to the speed of light?\n\nIf the speeds are very slow, the Schwarzschild solution (I.e. the metric tensor) is approximately \\[ds^2\\approx -\\phi dt^2 + dx^2 + dy^2 + dz^2\\]where \\[\\phi = c^2-\\frac{2GM}{r}\\] with \\(M\\) the mass of the sun and \\[r^2=x^2+y^2+z^2.\\]The geodesics parameterized by t give the equations of motion for \\(x,y,z\\) as \\[\\frac{d^2x}{dt^2}=\\frac{\\partial}{\\partial x}\\]and likewise for \\(y\\) and \\(z\\). This is exactly the equations of motion from Newton’s law of gravity due to the gravitational field of a point mass. \n\nIs this more or less correct?",
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